All airlines (although see note about TG) appear to give them departing from Bangkok, so the list below is mostly useful for inbound. It's important to remember that this is Thailand so just because an airline has given them out before is no guide they'll be given out in your case. It's worth asking both air-crew and the ground crew at the gate for them, as they're also occasionally forgotten. Thus the list below tends to reflect "this airline has given them out at least once" rather than being any guide to what will actually happen...

1) Immigration seems to only care about the airline stamp. Recent experience continues to be hassle free when the printed date and/or names do not match or are not filled in. YMMV. Edit: In early June (2015) was told by dragon to fill in the name field on the Fast Track voucher. She then checked to see if it matched BP name.

2) If you're inbound to BKK and have not received a voucher onboard the aircraft, be sure to ask!

3) Fast track vouchers do not have to be used on the date issued. If regular immigration queues are short on your arrival date, it's often worthwhile to pocket the voucher for a different arrival date when you may not have premium access (or to give as a gift to friends/family coming to visit).

EK provides fast track on departure and arrival for business and first class.
Arrival fast track cards are EK cards without any date on it.
Departure are BKK Airport fast trackcards with name and date written on it.

Any idea if I can convince UA to give me one in Y? I got downgraded from C due to irrops.

My hotel offers Fast Track passes, but charge 1500 THB for them (!). I'd prefer not to fork out $50 for something the airline had previously been giving me for free.

I assume you are talking about departure,

I was in the First Class check-in and the lady did not even know they had them, I asked and the other First Class check in lady told her in Thai where they were,

I would bet that if you check in using the coach line that they will not have them......you can always ask, or like someone else said there are other immigration gates you might want to look for them.....

Flying on TG in C from PEK, any experience getting them from TG or will I need to ask?

If flying TG F or C you don't need a pass, your boarding pass will suffice. However, they frequently ask to see the boarding pass and I have seen some passengers get to the front of the line only to get redirected to the regular lines by the immigration officer.

Just flew out of BKK the other day on UA and they gave me one. I assumed it was because I was running a little bit late. It ended up being no faster than the regular immigration line as there were only two agents working the fast track immigration lines.